TOPICS
Search

Chebyshev Approximation Formula

Using a Chebyshev polynomial of the first kind T(x), define

c_j=2/Nsum_(k=1)^(N)f(x_k)T_j(x_k)
(1)
=2/Nsum_(k=1)^(N)f[cos{(pi(k-1/2))/N}]cos{(pij(k-1/2))/N}.
(2)

Then

f(x) approx sum_(k=0)^(N-1)c_kT_k(x)-1/2c_0.
(3)

It is exact for the N zeros of T_N(x). This type of approximation is important because, when truncated, the error is spread smoothly over [-1,1]. The Chebyshev approximation formula is very close to the minimax polynomial.

Explore with Wolfram|Alpha

References

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. "Chebyshev Approximation," "Derivatives or Integrals of a Chebyshev-Approximated Function," and "Polynomial Approximation from Chebyshev Coefficients." §5.8, 5.9, and 5.10 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 184-188, 189-190, and 191-192, 1992.

Referenced on Wolfram|Alpha

Chebyshev Approximation Formula

Cite this as:

Weisstein, Eric W. "Chebyshev Approximation Formula." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ChebyshevApproximationFormula.html

Subject classifications